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# Complex Number

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 Sub Topics The complex numbers definition can be given as any number which can be represented in the form: x + iy, where ‘x’ is known as the real part of the complex number and iy is known as the imaginary part of the complex number. But you must know in complex number definition that x and y are both Real Numbers and i is the imaginary unit which satisfies the property i2 = -1 Let us have an example in complex numbers 2.5 + 2i is a complex number, we can also represent a real number a by simply writing a + 0i and for an Imaginary Number bi we can write 0 + bi. In complex number definition, we denote the Set of all the complex numbers by Z, the real part is denoted by Re(Z) or just R(z) and the imaginary part bi is denoted by Im(Z). In complex numbers definition, the complex numbers are being represented in their own plane called Complex Plane and the total idea of the two dimension complex plane has been extended from the one dimension number line where real part is represented on the horizontal axis and imaginary part on the vertical axis. In complex number definition, the number x + iy can also be identified by using a Point (x, y). One very important point in complex numbers definition is that when the real part is zero then that number is purely imaginary whereas when the imaginary part is zero then that number is purely real number. In complex number definition, when we have a negative imaginary part like 2 + (-5i), then we generally write it as 2 – 5i with b > 0 instead of writing as latter. There are many applications of complex numbers such as scientific fields, applied mathematics, quantum physics and engineering electromagnetism.

## Complex Plane

All complex numbers, which are represented in the form of a + bi are shown on the complex plane. It is also called as z- plane. It is used to represent the complex numbers established by the help of the real and the orthogonal – imaginary axis. Here if we represent the complex number along the real axis, we can represent the imaginary part along the y- axis. The plane so formed is called the Complex Number plane.
We use the concept of the complex plane which allows a geometric interpretation of complex numbers. Now let us look at the complex numbers. The numbers which are written in the form of ( a + ib), where we have a and b as the Real Numbers and I = root (-1) is the representation of the complex number. The Set of all complex numbers are represented by C. For any complex number, z = a + ib, we have a as the real part of z, which is written as Re( z) and b is the imaginary part of z, written as Im( z).
(4 + 3i), here 4 is real part of the complex number and 3 is the imaginary part of the complex number.
We say that the number is a pure real number, when we have the Im (z) = 0, such as 4 , 7, root(3) are all the pure real numbers. On the other hand, we have the pure Imaginary Number, if the real part of the complex number i.e. a = 0. Thus, we write Re (z) = 0 2i, root (7) I, -3i/2 are all pure imaginary numbers.
Modulus of the complex number z is represented as |z| = root ( a^2 + b^2).
Suppose we have a complex number z = 3 + 4 I, then the modulus of z = root ( 9 + 16) = root (25) = 5.